Journal of Global Optimization

, Volume 25, Issue 4, pp 407–423 | Cite as

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

  • Le Thi Hoai An
  • Pham Dinh Tao
  • Dinh Nho Hào


An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.

branch-and-bound technique DCA d.c. programming ill-posed problem inverse problem Tikhonov regularization 


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  1. 1.
    Bui, H. D. (1994), Inverse Problems in the Mechanics of Materials, CRC Press, Boca Raton, FL.Google Scholar
  2. 2.
    Chavent, G. and Kunisch, K. (1994) Convergence of Tikhonov regularization for constrained ill-posed inverse problems, Inverse Problems 10: 63-76.Google Scholar
  3. 3.
    Colonius, F. and Kunisch, K. (1986), Stability for parameter estimation in two point boundary value problem, J. Reine Angewandte Math. 370: 1-29.Google Scholar
  4. 4.
    Colonius, F. and Kunisch, K. (1989), Output least squares stability in elliptic systems, Appl. Math. Opt. 19: 33–63.Google Scholar
  5. 5.
    Engl, H. W., Hanke, M. and Neubauer, A. (1996), Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht.Google Scholar
  6. 6.
    Falk, J. E. and Soland, R. M. (1969), An algorithm for separable nonconvex programming problems, Management Science 15: 550-569.Google Scholar
  7. 7.
    Kalantari, B. and Rosen, J. B. (1987), Algorithm for global minimization of linearly constrained concave quadratic functions, Mathematics of Operations Research 12: 544-561.Google Scholar
  8. 8.
    Engl, H.W., Kunisch, K. and Neubauer, A. (1989), Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems 5: 523-540Google Scholar
  9. 9.
    Groetsch, Ch. W. (1993), Inverse Problems in the Mathematical Sciences. Friedr. Vieweg & Sohn, Braunschweig.Google Scholar
  10. 10.
    Horst, R. and Tuy, H. (1993), Global Optimization: Deterministic Approaches, 2 edition, Springer, Berlin, New York.Google Scholar
  11. 11.
    Isakov, V. (1998) Inverse Problems for Partial Differential Equations, Springer, New York.Google Scholar
  12. 12.
    Keller, J. B. (1976), Inverse problems, Amer. Math. Monthly 83: 107-118.Google Scholar
  13. 13.
    Kunisch, K. and Ring, W. (1993), Regularization of nonlinear ill-possed problems with closed operators, Numer. Funct. Anal. Optimiz. 14: 389-404.Google Scholar
  14. 14.
    Ladyzhenskaya, O. A. (1985), The Boundary Value Problems of Mathematical Physics, Springer, New York, Berlin, Heidelberg, Tokyo.Google Scholar
  15. 15.
    Le Thi Hoai An (1997), Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des Recherches, Université de Rouen.Google Scholar
  16. 16.
    Le Thi Hoai An and Pham Dinh Tao (1997), Solving a class of linearly constrained indefinite quadratic problems by d.c. algorithms, Journal of Global Optimization 11: 253-285.Google Scholar
  17. 17.
    Le Thi Hoai An and Pham Dinh Tao (1998), A branch-and-bound method via d.c. optimization algorithm and ellipsoidal technique for box constrained nonconvex quadratic programming problems, Journal of Global Optimization, 13: 171-206.Google Scholar
  18. 18.
    Le Thi Hoai An and Pham Dinh Tao (2001), The D.C. programming and DCA revisited with D.C. models of real world nonconvex optimization problems, Proceedings of The 5th International Conference on Optimization: Techniques and Applications, Hong Kong December 15-17, 2001 (ICOTA 2001) Editor D. Li, Volume 3, pp. 1324-1333.Google Scholar
  19. 19.
    Le Thi Hoai An and Pham Dinh Tao (2002), Large Scale Global Molecular Optimization from exact distance matrices by a d.c. optimization approach, Revised version in SIAM Journal on Optimization.Google Scholar
  20. 20.
    Le Thi Hoai An, Pham Dinh Tao, and Dinh Nho Hào (2001), On the ill-posedness of the trust region subproblem. Journal of Inverse and Ill-Posed Problems (to appear)Google Scholar
  21. 21.
    Le Thi Hoai An, Pham Dinh Tao, and Dinh Nho Hào (2001), D.c. programming approach to Tikhonov regularization for non-linear ill-posed problems, (in preparation).Google Scholar
  22. 22.
    Pham Dinh Tao and E. B. Souad (1986), Algorithms for solving a class of non convex optimization problems. Methods of subgradients, Fermat days 85. Mathematics for Optimization, Elsevier, North-Holland, Amsterdam: 249-270.Google Scholar
  23. 23.
    Pham Dinh Tao and E. B. Souad (1988) Duality in d. c. (difference of convex functions) optimization. Subgradient methods, Trends in Mathematical Optimization, International Series of Numer Math. 84 (Birkhäuser), 277-293.Google Scholar
  24. 24.
    Pham Dinh Tao and Le Thi Hoai An (1998), D.c. optimization algorithms for solving the trust region subproblem, SIAM J. Optimization 8: 476-505.Google Scholar
  25. 25.
    Pham Dinh Tao and Le Thi Hoai An (1997), Convex analysis approach to d.c. programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica 22(1): 289-355.Google Scholar
  26. 26.
    Polyak, B. (1987), Introduction to Optimization. (Optimization Software Inc., Publication Division, New York.Google Scholar
  27. 27.
    Rockafellar, R. T. (1970), Convex Analysis, Princeton University Press, Princeton.Google Scholar
  28. 28.
    Tikhonov, A. N., Leonov, A. S. and Yagola, A. G. (1998), Non-linear Ill-Posed Problems, Vol. 1, 2, Chapman and Hall, London.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Le Thi Hoai An
    • 1
  • Pham Dinh Tao
    • 1
  • Dinh Nho Hào
    • 1
    • 2
  1. 1.Laboratoire de Modélisation et Optimisation-Recherche Opérationnelle, BP 8Institut National des Sciences Appliquées de RouenMont Saint AignanFrance
  2. 2.Hanoi Institute of MathematicsHanoiVietnam

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